For all differentiation matrices, there is the question of accuracy of computation of the polynomial derivative by matrix multiplication. In general, differentiation is infinitely ill-conditioned: the derivative of \(f(x) + \varepsilon v(x)\)  can be arbitrarily different to the derivative of \(f(x)\) . However, if both \(f\)  and the perturbation are restricted to be polynomial, then the ill-conditioning is finite, and the absolute condition number is bounded by the norm of the differentiation matrix \(\mathbf {D}\) . This is Theorem 11.2 of [1]}, which we state formally below.

If \(f(x)\)  and \(\Delta f(x)\)  are both polynomials of degree at most \(n\) , and are both expressed in a polynomial basis \(\phi \) , then
\(\Vert  \Delta f^{\prime }(x) \Vert  \le \Vert  \mathbf {D}_{\phi } \Vert  \Vert  \Delta f (x) \Vert \) 
